The restrained rainbow bondage number of a graph

Main Article Content

Jafar Amjadi
Rana Khoeilar
N. Dehgardi
Lutz Volkmann
S.M. Sheikholeslami

Abstract

A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the restrained $k$-rainbow domination number of $G$, denoted by $\gamma_{rrk}(G)$. The restrained $k$-rainbow bondage number $b_{rrk}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $F \subseteq E(G)$ for which $\gamma_{rrk}(G-F) > \gamma_{rrk}(G)$. In this paper, we initiate the study of the restrained $k$-rainbow bondage number in graphs and we present some sharp bounds for $b_{rr2}(G)$. In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.

Article Details

How to Cite
Amjadi, J., Khoeilar, R., Dehgardi, N., Volkmann, L., & Sheikholeslami, S. (2018). The restrained rainbow bondage number of a graph. Tamkang Journal of Mathematics, 49(2), 115–127. https://doi.org/10.5556/j.tkjm.49.2018.2365
Section
Papers
Author Biographies

Jafar Amjadi

Department of Mathematics, Azarbaijan ShahidMadani University, Tabriz, I.R. Iran.

Rana Khoeilar

Department of Mathematics, Azarbaijan ShahidMadani University, Tabriz, I.R. Iran.

N. Dehgardi

Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, I.R. Iran.

Lutz Volkmann

Lehrstuhl II fürM athematik, RWTH Aachen University, 52056 Aachen, Germany.

S.M. Sheikholeslami

Department ofMathematics, Azarbaijan ShahidMadani University, Tabriz, I.R. Iran.

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